Relational bundle geometric formulation of non-relativistic quantum mechanics
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915936781467648 |
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| author | François, J. Ravera, L. |
| author_facet | François, J. Ravera, L. |
| contents | We present a bundle geometric formulation of non-relativistic many-particles Quantum Mechanics. A wave function is seen to be a $\mathbb{C}$-valued cocyclic tensorial 0-form on configuration space-time seen as a principal bundle, while the Schrödinger equation flows from its covariant derivative, with the action functional supplying a (flat) cocyclic connection 1-form on the configuration bundle. In line with the historical motivations of Dirac and Feynman, ours is thus a Lagrangian geometric formulation of QM, in which the Dirac-Feynman path integral arises in a geometrically natural way. Applying the dressing field method, we obtain a relational reformulation of this geometric non-relativistic QM: a relational wave function is realised as a basic cocyclic 0-form on the configuration bundle. In this relational QM, any particle position can be used as a dressing field, i.e. as a "physical reference frame". The dressing field method naturally accounts for the freedom in choosing the dressing field, which is readily understood as a covariance of the relational formulation under changes of physical reference frame. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02046 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relational bundle geometric formulation of non-relativistic quantum mechanics François, J. Ravera, L. Quantum Physics High Energy Physics - Theory Mathematical Physics We present a bundle geometric formulation of non-relativistic many-particles Quantum Mechanics. A wave function is seen to be a $\mathbb{C}$-valued cocyclic tensorial 0-form on configuration space-time seen as a principal bundle, while the Schrödinger equation flows from its covariant derivative, with the action functional supplying a (flat) cocyclic connection 1-form on the configuration bundle. In line with the historical motivations of Dirac and Feynman, ours is thus a Lagrangian geometric formulation of QM, in which the Dirac-Feynman path integral arises in a geometrically natural way. Applying the dressing field method, we obtain a relational reformulation of this geometric non-relativistic QM: a relational wave function is realised as a basic cocyclic 0-form on the configuration bundle. In this relational QM, any particle position can be used as a dressing field, i.e. as a "physical reference frame". The dressing field method naturally accounts for the freedom in choosing the dressing field, which is readily understood as a covariance of the relational formulation under changes of physical reference frame. |
| title | Relational bundle geometric formulation of non-relativistic quantum mechanics |
| topic | Quantum Physics High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2501.02046 |